Eventually periodic sequence

Tags:

Description

Given is a function

f: 0..N --> 0..N

for a non-negative

N

and a non-negative integer

n

≤

N

. One can construct an infinite sequence

F = f ¹(n), f ²(n), ... f ^k(n) ...

, where

f ^k(n)

is defined recursively as follows:

f ¹(n) = f(n)

and

f ^k+1(n)

=

f(f ^k(n))

.

It is easy to see that each such sequence F is eventually periodic, that is periodic from some point onwards, e.g 1, 2, 7, 5, 4, 6, 5, 4, 6, 5, 4, 6 ... . Given non-negative integer N ≤ 11000000 , n ≤ N and f, you are to compute the period of sequence F.

Each line of input contains N, n and the a description of f in postfix notation, also known as Reverse Polish Notation (RPN). The operands are either unsigned integer constants or N or the variable x. Only binary operands are allowed: + (addition), * (multiplication) and % (modulo, i.e. remainder of integer division). Operands and operators are separated by whitespace. The operand % occurs exactly once in a function and it is the last (rightmost, or topmost if you wish) operator and its second operand is always N whose value is read from input. The following function:

 
            2 x * 7 + N % 

is the RPN rendition of the more familiar infix

(2*x+7)%N

. All input lines are shorter than 100 characters. The last line of input has

N

equal 0 and should not be processed.

For each line of input, output one line with one integer number, the period of F corresponding to the data given in the input line.

Samples